Problem

Simplify (4x^3y^-3)/(2x^3x^-1y^2)

The question is asking for the simplification of a given algebraic expression involving variables raised to exponent powers. Specifically, you need to simplify the fraction where both the numerator and the denominator contain terms with variables x and y raised to different powers. The process likely involves applying the rules of exponents, such as the product of powers rule, the quotient of powers rule, and the negative exponent rule, to combine and reduce the expression to its simplest form.

$\frac{4 x^{3} y^{- 3}}{2 x^{3} x^{- 1} y^{2}}$

Answer

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Solution:

Step 1: Combine $x$ terms in the denominator

  • Identify powers of $x$ to be combined: $\frac{4x^3y^{-3}}{2(x^{-1}x^3)y^2}$
  • Apply the exponent addition rule: $a^{m}a^{n} = a^{m+n}$
  • Simplify the $x$ terms: $\frac{4x^3y^{-3}}{2x^{3-1}y^2}$

Step 2: Simplify negative exponent for $y$

  • Apply the negative exponent rule: $b^{-n} = \frac{1}{b^n}$
  • Move $y^{-3}$ to the denominator: $\frac{4x^3}{2x^2y^2y^3}$

Step 3: Combine $y$ terms in the denominator

  • Identify powers of $y$ to be combined: $\frac{4x^3}{2x^2(y^3y^2)}$
  • Apply the exponent addition rule: $a^{m}a^{n} = a^{m+n}$
  • Simplify the $y$ terms: $\frac{4x^3}{2x^2y^{3+2}}$

Step 4: Reduce the fraction by cancelling common factors

  • Identify common factors in numerator and denominator: $4$ and $2$
  • Factor out the common factor of $2$: $\frac{2(2x^3)}{2(x^2y^5)}$
  • Cancel out the common factor of $2$: $\frac{2x^3}{x^2y^5}$

Step 5: Cancel common $x$ terms

  • Identify common $x$ factors: $x^3$ and $x^2$
  • Factor out the common $x^2$: $\frac{x^2(2x)}{x^2y^5}$
  • Cancel out the common $x^2$: $\frac{2x}{y^5}$

Knowledge Notes:

  • Exponent addition rule: When multiplying like bases, add the exponents: $a^{m}a^{n} = a^{m+n}$.

  • Negative exponent rule: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent: $b^{-n} = \frac{1}{b^n}$.

  • Simplifying fractions: To simplify a fraction, cancel out common factors in the numerator and the denominator.

  • Combining like terms: Terms with the same base and exponent can be combined by adding or subtracting the coefficients.

  • Reducing fractions: To reduce a fraction to its simplest form, divide both the numerator and the denominator by their greatest common factor (GCF).

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